BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI K K
BIRLA GOA CAMPUS
SECOND SEMESTER 2016-2017
Thursday, 4th May, 2017
Comprehensive Examination
(Closed Book)
Course Title: MATHEMATICS-III Max. Marks: 120
Course No: MATH F 211 Time: 3 hour
Instructions
(i) Answer all questions. Start a new question in a new page and answer all its
parts in the same place.
(ii) Write all the steps clearly and give explanations for the complete credit.
(iii) Make an index on the front page of the main answer sheet. Otherwise 2
marks will be deducted.
dy x+y+4
1. Solve dx
= x−y−4
. [10]
2 +y 2 2 +y 2
2. Solve xex dx + y(ex + 1)dy = 0 with y(0) = 0. [05]
∂M
− ∂N R
3. If ∂y N ∂x = f (x), a function of x alone, then prove that e f (x)dx is an integrating factor
of the differential equation M (x, y)dx + N (x, y)dy = 0. [05]
d2 y 1 dy 2
4. Find general solution of the differential equation dx 2 + 1 + y dx
= 0. [10]
5. Consider the following differential equation [15]
00 0
(x2 − 2x)y − (x2 − 2)y + 2(x − 1)y = 3x2 (x − 2)2 ex , x 6= 0, 2.
If y(x) = ex is one of the solution of corresponding homogeneous differential equation,
then find second. Also, find the particular solution of the above differential equation
by using method of variation parameters and hence, write the general solution.
6. If Jp (x) is the Bessel function of first kind then, prove that [15]
000
8Jn = Jn−3 − 3Jn−1 + 3Jn+1 − Jn+3
000 0
and use it to show that 8J0 + 2J3 + 6J0 = 0.
7. The equation for one dimensional transverse vibration of the stretched string is given
2 ∂2y
by ∂∂t2y = a2 ∂x2 , where a > 0 is constant. Find the solution y(x, t) using method of
separation of variable with initial conditions [15]
(
2x
, 0 ≤ x ≤ π2 ∂y
y(x, 0) = π(π−x) π , |t = 0
2 π , 2 ≤x≤π ∂t
and boundary conditions
y(0, t) = y(π, t) = 0.
[15]
1
8. Find the Fourier series of the periodic function dfined by [15]
(
0, − π ≤ x < 0;
f (x) =
x2 , 0 ≤ x < π.
and use the Fourier series to obtain the sums
1 1 1 π2
1+ + + + . . . = .
22 32 42 6
9. Solve the initial value problem using laplace transform [15]
0, 0 ≤ t < 1;
00 0 0
y − 2y + y = 1, 1 ≤ t < 2; , y(0) = 0; y (0) = 1.
0, 2 ≤ t < ∞
10. Find the power series solutions in powers of x − 1 of the following initial value problem:
00 0 0
x(2 − x)y − 6(x − 1)y − 4y = 0; y(1) = 0, y (1) = 0.
[15]
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